CAPITULO 3. IMPLEMENTACIÓN DEL MODELO EN UN CASO DE ESTUDIO
3.2 FASE 2: PLANIFICACIÓN
3.2.1 PLAN DEL ALCANCE
In elementary accounts o f group theory [2], it is usual to express symmetry operations in terms o f matrices. In this way the matrix equation
y' =
V
sin 8 cos# 0
0 0 U
(3.2.1)
represents the effect on a point (xyz) o f a rotation o f the space through an angle 0 around the z axis. It is moved to a new point (x’ y’ z' ) . The 3x3 matrix can be taken to stand for the symmetry operation referred to a set o f Cartesian coordinates chosen in this particular way. It is an orthogonal matrix i.e. in general its (real) elements aij satisfy the equations
(3.2.2)
and the determinant o f its elements is ±1. Equation (2) implies that the inverse o f an orthogonal matrix is equal to its transpose.
An orthogonal matrix is a special example o f a unitary matrix, whose determinant is
also ±1 but whose (complex) elements satisfy
(3.2.3)
k
The inverse o f a unitary matrix is the transpose o f its complex conjugate matrix. In this way an orthogonal matrix is a unitary matrix whose elements are all real.
Equation (1) may be written in the more compact general form:
r’= A r (3.2.4)
The particular entries in the matrix A depend on the choice o f coordinate axis directions. A change in the directions gives rise to an equivalent matrix. Thus if s = Tr
in which ^ is the vector written with respect to axes transformed by the orthogonal
matrix T s’= Bs where B = TAT'^
is equivalent to A (i.e. describes the same symmetry operations as A) with respect to the new coordinate axes.
Certain properties o f the symmetry operation are invariant under the transformation, in
particular the value o f the determinant and the character
%
i.e. the sum o f the diagonalelements. In this way for the example given X = Z a j = Z b j j
j j
Again, referring to the three dimensional point groups, it is well known that all symmetry operations can be classified as rotations (C in the Schonfiies notation) and rotation-refiections (S). The pure rotations (C) are characterised by determinant +1 and the rotation-refiections (S) by determinant -1. Symmetry operations in the general many dimensional case can be classified according to the value o f the determinant (+1 or -1) and can be described using the terms rotation and reflection.
The treatment [12, 13] begins by recalling that an orthogonal matrix A always has an equivalent diagonal matrix, the elements o f which are the distinct roots o f the equation IA - A,l| = 0
where I is the identity matrix. The vertical bars indicate the determinant o f the matrix
so that the equation represents a polynomial o f order n in the unknown
X .
For the matrix given in (1) the three roots are found to be
+1, e’®, e ’®
The occurrence o f complex quantities indicates that the diagonal matrix is unitary, i.e. its elements satisfy equation (3).
Correspondingly, the transformation matrix T to the diagonal matrix A A = T A T '
becomes unitary.
The pair o f values e'® and e''® is diagnostic o f a rotation through 0 in a plane (in this case the xy plane). The diagonalised form separates complex axes (x+iy) and (x-iy). These ideas can be generalised. If an orthogonal matrix o f any dimensionality is diagonalised, the diagonal elements may only take the following values
+1 ; -1 ; or the pair e'® and e*'®
The determinant o f a diagonal matrix is clearly the product o f the diagonal elements. The value o f any combination o f the permitted elements is ±1 as required for a unitary matrix.
Each o f the permitted diagonal elements can be given a simple geometrical interpretation which persists however high the dimensionality o f the space being considered.
The value +1 corresponds to a (real) direction in the space which is left unchanged by the symmetry operation. It will be referred to as an identity (direction).
The value -1 corresponds to a (real) direction which is reversed in sign by the symmetry operation. In three dimensional language it is conventional to speak o f a
reflection plane. This is a passive convention emphasising the entity which is left unchanged. It is equally permissible to adopt an active convention and refer to a reflection line: the line in three dimensions being normal to the plane and reversed in sign. This convention can be used to define a reflection in any number o f dimensions. A reflection line refers to a single direction which is altered in sign. In a pure reflection, the (n-1) dimensional subspace orthogonal to the line is left unchanged. In this way the value -1 will correspond to a reflection (direction).
It has been seen that the complex conjugate pair e‘^ and e''^ is associated with rotation through 0. Again following an active convention, in three dimensions it is emphasised that rotation takes place in a plane (as opposed to round an axis). A generalised rotation in any number o f dimensions refers to a plane and in a pure rotation the (n-2) dimensional subspace orthogonal to the plane is left unchanged. A pair o f eigenvalues e'^ e^^ is then diagnostic o f a rotation plane.
Since these are the only possibilities a generalised notation using (identity) reflection and rotation only can be built up. Operations are divided according to the value o f the determinant +1 or -1.
Beginning in 1 -dimension, there are only tw o possibilities
+1 -1
Identity Reflection
In two dimensions, determinant +1 is achieved by the pair e'®,e’’® (a rotation). Determinant -1 can only be maintained by adding +1
e'^ e"^ +1 -1
In three dimensions, determinant +1 can only be maintained by adding +1, determinant -1 by replacing +1 by the pair e’^,e*'^
+1 e '' e"' e '' e"' -1
(Identity)-rotation Rotation-reflection
In four dimensions, determinant +1 is maintained by replacing +1 by e'* e''* and -1 can only be maintained by adding +1
e‘® e ’® e'* e'* +I e‘® e ’® -1
Rotation-rotation (Identity)-rotation-reflection
Higher dimensionalities are reached by repeating the procedure i.e. either a) adding identity if it does not occur
b) replacing identity by rotation if it does occur.
It will be seen that only one essentially new symmetry type is produced for each
dimension added. All +1 symmetry operations are compounded from rotations (and the identity) only. All -1 symmetry operations always contain one reflection direction. The compound rotation-rotation symmetry element type which appears in four dimensions can be pictured in a relatively straight forward way. Just as in two dimensions, two orthogonal lines meet only in one point, so in four dimensions, two orthogonal planes intersect only in one point. The implication is that independent
rotations (indicated by independent values o f the angles
^
and 0) can take place in thetwo planes. All +1 symmetry operations in four dimensions can be expressed in this way.
The classification just given is compact in the sense that, for any dimension, symmetry operations can be allocated to one o f only two types. Particular examples occur which have special significance. Thus in three dimensions, a rotation-reflection through 0
reduces to the three diagonal elements +1 +1 -1. This is identifiable as a refiection line and is given the special symbol a in the Schonfiies notation. On the other hand, a
rotation-reflection through ti reduces t o-1 -1 -1. This is the familiar inversion (denoted
by /). The point is that there is now an ambiguity in the interpretation o f the rotation- reflection. Is the rotation in the xy plane and the reflection along z or is the rotation in the xz plane and the reflection along y? Clearly there is no physical distinction between these two possibilities. Similar, but much less familiar, special cases arise in higher dimensions. Thus, in four dimensions, for a rotation-rotation the rotation angle in one o f the planes may be 0 leading to the diagonal elements +1 +1 e'® e"'®. This special case shows that the lower dimensional possibility o f a simple rotation is implicitly contained in the four dimensional classification. This is always the case: all lower dimensional operations are contained in higher order type.
A second special case corresponds to rotations o f ti in both planes leading to the diagonal values -1 -1 -1 -1. This is the four dimensional analogue o f the inversion and
contains a similar ambiguity. Is the operation a Tt rotation in xy together with a 7t
rotation in zw or a Tt rotation in the xz plane together with a
n
rotation in the ywplane? Once again there is no physical distinction. It might be anticipated that point groups which are peculiarly four dimensional may reflect this ambiguity.
A more subtle ambiguity arises when the two angles in a double rotation are equal so
that the elements o f the equivalent diagonal matrix become e'® e'® e'*® and the
repeated roots indicate that the choice o f underlying equivalent eigenvector is not uniquely determined.
If attention is restricted to orthogonal matrices, only a partial diagonalisation is possible giving the following form
rc -s 0 0"
s c 0 0
0 0 0 -s
.0 0 s C;
in which c = cos0 and s = sin0.
If attention is focussed on the upper diagonal block the resulting 2 x 2 matrix describes a rotation through 0 in the xy plane. It is noted that the matrix is independent o f the precise pair o f orthogonal directions chosen for x and y i.e. the rotation matrix is invariant under a rotation o f the Cartesian axis directions. M oreover in the four dimensional case, this invariance remains whether the rotation angle in the zw plane is the same or not. The axes in the zw plane may o f couse also be rotated independently without changing the matrix.
It follows that any additional invariances which result from the equi-angular double rotation can be expressed as a coordinate rotation in the plane xz plus a coordinate rotation in yw. It can be shown that invariance is only maintained if the rotation angles are the same. In terms o f a matrix equation